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The real part of a holomorphic function is harmonic
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Anthony QuasJan 15 '11 at 19:05

I think that any book on complex analysis (or several complex variables) will discuss the connection at great length, so there is no need to write essays, long or otherwise.
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Igor RivinJan 15 '11 at 22:26

1

See the paper Harmonic functions from a complex analysis viewpoint, by Sheldon Axler. (Google it to find an online version.)
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Charles StaatsJan 16 '11 at 0:51

If I am not mistaken, the construction of the harmonic conjugate of a given harmonic function works locally, so simple connectedness only becomes an issue on a larger scale.
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drbobmeisterJan 20 '11 at 4:37